# Automatic parallelization with `@jit`

¶

Setting the parallel option for `jit()`

enables
a Numba transformation pass that attempts to automatically parallelize and
perform other optimizations on (part of) a function. At the moment, this
feature only works on CPUs.

Some operations inside a user defined function, e.g. adding a scalar value to
an array, are known to have parallel semantics. A user program may contain
many such operations and while each operation could be parallelized
individually, such an approach often has lackluster performance due to poor
cache behavior. Instead, with auto-parallelization, Numba attempts to
identify such operations in a user program, and fuse adjacent ones together,
to form one or more kernels that are automatically run in parallel.
The process is fully automated without modifications to the user program,
which is in contrast to Numba’s `vectorize()`

or
`guvectorize()`

mechanism, where manual effort is required
to create parallel kernels.

## Supported Operations¶

In this section, we give a list of all the array operations that have parallel semantics and for which we attempt to parallelize.

All numba array operations that are supported by Case study: Array Expressions, which include common arithmetic functions between Numpy arrays, and between arrays and scalars, as well as Numpy ufuncs. They are often called element-wise or point-wise array operations:

unary operators:

`+`

`-`

`~`

binary operators:

`+`

`-`

`*`

`/`

`/?`

`%`

`|`

`>>`

`^`

`<<`

`&`

`**`

`//`

comparison operators:

`==`

`!=`

`<`

`<=`

`>`

`>=`

Numpy ufuncs that are supported in nopython mode.

User defined

`DUFunc`

through`vectorize()`

.

Numpy reduction functions

`sum`

,`prod`

,`min`

,`max`

,`argmin`

, and`argmax`

. Also, array math functions`mean`

,`var`

, and`std`

.Numpy array creation functions

`zeros`

,`ones`

,`arange`

,`linspace`

, and several random functions (rand, randn, ranf, random_sample, sample, random, standard_normal, chisquare, weibull, power, geometric, exponential, poisson, rayleigh, normal, uniform, beta, binomial, f, gamma, lognormal, laplace, randint, triangular).Numpy

`dot`

function between a matrix and a vector, or two vectors. In all other cases, Numba’s default implementation is used.Multi-dimensional arrays are also supported for the above operations when operands have matching dimension and size. The full semantics of Numpy broadcast between arrays with mixed dimensionality or size is not supported, nor is the reduction across a selected dimension.

Array assignment in which the target is an array selection using a slice or a boolean array, and the value being assigned is either a scalar or another selection where the slice range or bitarray are inferred to be compatible.

The

`reduce`

operator of`functools`

is supported for specifying parallel reductions on 1D Numpy arrays but the initial value argument is mandatory.

## Explicit Parallel Loops¶

Another feature of the code transformation pass (when `parallel=True`

) is
support for explicit parallel loops. One can use Numba’s `prange`

instead of
`range`

to specify that a loop can be parallelized. The user is required to
make sure that the loop does not have cross iteration dependencies except for
supported reductions.

A reduction is inferred automatically if a variable is updated by a binary
function/operator using its previous value in the loop body. The initial value
of the reduction is inferred automatically for the `+=`

, `-=`

, `*=`

,
and `/=`

operators.
For other functions/operators, the reduction variable should hold the identity
value right before entering the `prange`

loop. Reductions in this manner
are supported for scalars and for arrays of arbitrary dimensions.

The example below demonstrates a parallel loop with a
reduction (`A`

is a one-dimensional Numpy array):

```
from numba import njit, prange
@njit(parallel=True)
def prange_test(A):
s = 0
# Without "parallel=True" in the jit-decorator
# the prange statement is equivalent to range
for i in prange(A.shape[0]):
s += A[i]
return s
```

The following example demonstrates a product reduction on a two-dimensional array:

```
from numba import njit, prange
import numpy as np
@njit(parallel=True)
def two_d_array_reduction_prod(n):
shp = (13, 17)
result1 = 2 * np.ones(shp, np.int_)
tmp = 2 * np.ones_like(result1)
for i in prange(n):
result1 *= tmp
return result1
```

Care should be taken, however, when reducing into slices or elements of an array if the elements specified by the slice or index are written to simultaneously by multiple parallel threads. The compiler may not detect such cases and then a race condition would occur.

The following example demonstrates such a case where a race condition in the execution of the parallel for-loop results in an incorrect return value:

```
from numba import njit, prange
import numpy as np
@njit(parallel=True)
def prange_wrong_result(x):
n = x.shape[0]
y = np.zeros(4)
for i in prange(n):
# accumulating into the same element of `y` from different
# parallel iterations of the loop results in a race condition
y[:] += x[i]
return y
```

as does the following example where the accumulating element is explicitly specified:

```
from numba import njit, prange
import numpy as np
@njit(parallel=True)
def prange_wrong_result(x):
n = x.shape[0]
y = np.zeros(4)
for i in prange(n):
# accumulating into the same element of `y` from different
# parallel iterations of the loop results in a race condition
y[i % 4] += x[i]
return y
```

whereas performing a whole array reduction is fine:

```
from numba import njit, prange
import numpy as np
@njit(parallel=True)
def prange_ok_result_whole_arr(x):
n = x.shape[0]
y = np.zeros(4)
for i in prange(n):
y += x[i]
return y
```

as is creating a slice reference outside of the parallel reduction loop:

```
from numba import njit, prange
import numpy as np
@njit(parallel=True)
def prange_ok_result_outer_slice(x):
n = x.shape[0]
y = np.zeros(4)
z = y[:]
for i in prange(n):
z += x[i]
return y
```

## Examples¶

In this section, we give an example of how this feature helps parallelize Logistic Regression:

```
@numba.jit(nopython=True, parallel=True)
def logistic_regression(Y, X, w, iterations):
for i in range(iterations):
w -= np.dot(((1.0 / (1.0 + np.exp(-Y * np.dot(X, w))) - 1.0) * Y), X)
return w
```

We will not discuss details of the algorithm, but instead focus on how this program behaves with auto-parallelization:

Input

`Y`

is a vector of size`N`

,`X`

is an`N x D`

matrix, and`w`

is a vector of size`D`

.The function body is an iterative loop that updates variable

`w`

. The loop body consists of a sequence of vector and matrix operations.The inner

`dot`

operation produces a vector of size`N`

, followed by a sequence of arithmetic operations either between a scalar and vector of size`N`

, or two vectors both of size`N`

.The outer

`dot`

produces a vector of size`D`

, followed by an inplace array subtraction on variable`w`

.With auto-parallelization, all operations that produce array of size

`N`

are fused together to become a single parallel kernel. This includes the inner`dot`

operation and all point-wise array operations following it.The outer

`dot`

operation produces a result array of different dimension, and is not fused with the above kernel.

Here, the only thing required to take advantage of parallel hardware is to set
the parallel option for `jit()`

, with no
modifications to the `logistic_regression`

function itself. If we were to
give an equivalence parallel implementation using `guvectorize()`

,
it would require a pervasive change that rewrites the code to extract kernel
computation that can be parallelized, which was both tedious and challenging.

## Diagnostics¶

Note

At present not all parallel transforms and functions can be tracked through the code generation process. Occasionally diagnostics about some loops or transforms may be missing.

The parallel option for `jit()`

can produce
diagnostic information about the transforms undertaken in automatically
parallelizing the decorated code. This information can be accessed in two ways,
the first is by setting the environment variable
`NUMBA_PARALLEL_DIAGNOSTICS`

, the second is by calling
`parallel_diagnostics()`

, both methods give the same information
and print to `STDOUT`

. The level of verbosity in the diagnostic information is
controlled by an integer argument of value between 1 and 4 inclusive, 1 being
the least verbose and 4 the most. For example:

```
@njit(parallel=True)
def test(x):
n = x.shape[0]
a = np.sin(x)
b = np.cos(a * a)
acc = 0
for i in prange(n - 2):
for j in prange(n - 1):
acc += b[i] + b[j + 1]
return acc
test(np.arange(10))
test.parallel_diagnostics(level=4)
```

produces:

```
================================================================================
======= Parallel Accelerator Optimizing: Function test, example.py (4) =======
================================================================================
Parallel loop listing for Function test, example.py (4)
--------------------------------------|loop #ID
@njit(parallel=True) |
def test(x): |
n = x.shape[0] |
a = np.sin(x)---------------------| #0
b = np.cos(a * a)-----------------| #1
acc = 0 |
for i in prange(n - 2):-----------| #3
for j in prange(n - 1):-------| #2
acc += b[i] + b[j + 1] |
return acc |
--------------------------------- Fusing loops ---------------------------------
Attempting fusion of parallel loops (combines loops with similar properties)...
Trying to fuse loops #0 and #1:
- fusion succeeded: parallel for-loop #1 is fused into for-loop #0.
Trying to fuse loops #0 and #3:
- fusion failed: loop dimension mismatched in axis 0. slice(0, x_size0.1, 1)
!= slice(0, $40.4, 1)
----------------------------- Before Optimization ------------------------------
Parallel region 0:
+--0 (parallel)
+--1 (parallel)
Parallel region 1:
+--3 (parallel)
+--2 (parallel)
--------------------------------------------------------------------------------
------------------------------ After Optimization ------------------------------
Parallel region 0:
+--0 (parallel, fused with loop(s): 1)
Parallel region 1:
+--3 (parallel)
+--2 (serial)
Parallel region 0 (loop #0) had 1 loop(s) fused.
Parallel region 1 (loop #3) had 0 loop(s) fused and 1 loop(s) serialized as part
of the larger parallel loop (#3).
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
---------------------------Loop invariant code motion---------------------------
Instruction hoisting:
loop #0:
Failed to hoist the following:
dependency: $arg_out_var.10 = getitem(value=x, index=$parfor__index_5.99)
dependency: $0.6.11 = getattr(value=$0.5, attr=sin)
dependency: $expr_out_var.9 = call $0.6.11($arg_out_var.10, func=$0.6.11, args=[Var($arg_out_var.10, example.py (7))], kws=(), vararg=None)
dependency: $arg_out_var.17 = $expr_out_var.9 * $expr_out_var.9
dependency: $0.10.20 = getattr(value=$0.9, attr=cos)
dependency: $expr_out_var.16 = call $0.10.20($arg_out_var.17, func=$0.10.20, args=[Var($arg_out_var.17, example.py (8))], kws=(), vararg=None)
loop #3:
Has the following hoisted:
$const58.3 = const(int, 1)
$58.4 = _n_23 - $const58.3
--------------------------------------------------------------------------------
```

To aid users unfamiliar with the transforms undertaken when the parallel option is used, and to assist in the understanding of the subsequent sections, the following definitions are provided:

- Loop fusion
Loop fusion is a technique whereby loops with equivalent bounds may be combined under certain conditions to produce a loop with a larger body (aiming to improve data locality).

- Loop serialization
Loop serialization occurs when any number of

`prange`

driven loops are present inside another`prange`

driven loop. In this case the outermost of all the`prange`

loops executes in parallel and any inner`prange`

loops (nested or otherwise) are treated as standard`range`

based loops. Essentially, nested parallelism does not occur.

- Loop invariant code motion
Loop invariant code motion is an optimization technique that analyses a loop to look for statements that can be moved outside the loop body without changing the result of executing the loop, these statements are then “hoisted” out of the loop to save repeated computation.

- Allocation hoisting
Allocation hoisting is a specialized case of loop invariant code motion that is possible due to the design of some common NumPy allocation methods. Explanation of this technique is best driven by an example:

@njit(parallel=True) def test(n): for i in prange(n): temp = np.zeros((50, 50)) # <--- Allocate a temporary array with np.zeros() for j in range(50): temp[j, j] = i # ...do something with temp

internally, this is transformed to approximately the following:

@njit(parallel=True) def test(n): for i in prange(n): temp = np.empty((50, 50)) # <--- np.zeros() is rewritten as np.empty() temp[:] = 0 # <--- and then a zero initialisation for j in range(50): temp[j, j] = i # ...do something with temp

then after hoisting:

@njit(parallel=True) def test(n): temp = np.empty((50, 50)) # <--- allocation is hoisted as a loop invariant as `np.empty` is considered pure for i in prange(n): temp[:] = 0 # <--- this remains as assignment is a side effect for j in range(50): temp[j, j] = i # ...do something with temp

it can be seen that the

`np.zeros`

allocation is split into an allocation and an assignment, and then the allocation is hoisted out of the loop in`i`

, this producing more efficient code as the allocation only occurs once.

### The parallel diagnostics report sections¶

The report is split into the following sections:

- Code annotation
This is the first section and contains the source code of the decorated function with loops that have parallel semantics identified and enumerated. The

`loop #ID`

column on the right of the source code lines up with identified parallel loops. From the example,`#0`

is`np.sin`

,`#1`

is`np.cos`

and`#2`

and`#3`

are`prange()`

:Parallel loop listing for Function test, example.py (4) --------------------------------------|loop #ID @njit(parallel=True) | def test(x): | n = x.shape[0] | a = np.sin(x)---------------------| #0 b = np.cos(a * a)-----------------| #1 acc = 0 | for i in prange(n - 2):-----------| #3 for j in prange(n - 1):-------| #2 acc += b[i] + b[j + 1] | return acc |

It is worth noting that the loop IDs are enumerated in the order they are discovered which is not necessarily the same order as present in the source. Further, it should also be noted that the parallel transforms use a static counter for loop ID indexing. As a consequence it is possible for the loop ID index to not start at 0 due to use of the same counter for internal optimizations/transforms taking place that are invisible to the user.

- Fusing loops
This section describes the attempts made at fusing discovered loops noting which succeeded and which failed. In the case of failure to fuse a reason is given (e.g. dependency on other data). From the example:

--------------------------------- Fusing loops --------------------------------- Attempting fusion of parallel loops (combines loops with similar properties)... Trying to fuse loops #0 and #1: - fusion succeeded: parallel for-loop #1 is fused into for-loop #0. Trying to fuse loops #0 and #3: - fusion failed: loop dimension mismatched in axis 0. slice(0, x_size0.1, 1) != slice(0, $40.4, 1)

It can be seen that fusion of loops

`#0`

and`#1`

was attempted and this succeeded (both are based on the same dimensions of`x`

). Following the successful fusion of`#0`

and`#1`

, fusion was attempted between`#0`

(now including the fused`#1`

loop) and`#3`

. This fusion failed because there is a loop dimension mismatch,`#0`

is size`x.shape`

whereas`#3`

is size`x.shape[0] - 2`

.

- Before Optimization
This section shows the structure of the parallel regions in the code before any optimization has taken place, but with loops associated with their final parallel region (this is to make before/after optimization output directly comparable). Multiple parallel regions may exist if there are loops which cannot be fused, in this case code within each region will execute in parallel, but each parallel region will run sequentially. From the example:

Parallel region 0: +--0 (parallel) +--1 (parallel) Parallel region 1: +--3 (parallel) +--2 (parallel)

As alluded to by the Fusing loops section, there are necessarily two parallel regions in the code. The first contains loops

`#0`

and`#1`

, the second contains`#3`

and`#2`

, all loops are marked`parallel`

as no optimization has taken place yet.

- After Optimization
This section shows the structure of the parallel regions in the code after optimization has taken place. Again, parallel regions are enumerated with their corresponding loops but this time loops which are fused or serialized are noted and a summary is presented. From the example:

Parallel region 0: +--0 (parallel, fused with loop(s): 1) Parallel region 1: +--3 (parallel) +--2 (serial) Parallel region 0 (loop #0) had 1 loop(s) fused. Parallel region 1 (loop #3) had 0 loop(s) fused and 1 loop(s) serialized as part of the larger parallel loop (#3).

It can be noted that parallel region 0 contains loop

`#0`

and, as seen in the fusing loops section, loop`#1`

is fused into loop`#0`

. It can also be noted that parallel region 1 contains loop`#3`

and that loop`#2`

(the inner`prange()`

) has been serialized for execution in the body of loop`#3`

.

- Loop invariant code motion
This section shows for each loop, after optimization has occurred:

the instructions that failed to be hoisted and the reason for failure (dependency/impure).

the instructions that were hoisted.

any allocation hoisting that may have occurred.

From the example:

Instruction hoisting: loop #0: Failed to hoist the following: dependency: $arg_out_var.10 = getitem(value=x, index=$parfor__index_5.99) dependency: $0.6.11 = getattr(value=$0.5, attr=sin) dependency: $expr_out_var.9 = call $0.6.11($arg_out_var.10, func=$0.6.11, args=[Var($arg_out_var.10, example.py (7))], kws=(), vararg=None) dependency: $arg_out_var.17 = $expr_out_var.9 * $expr_out_var.9 dependency: $0.10.20 = getattr(value=$0.9, attr=cos) dependency: $expr_out_var.16 = call $0.10.20($arg_out_var.17, func=$0.10.20, args=[Var($arg_out_var.17, example.py (8))], kws=(), vararg=None) loop #3: Has the following hoisted: $const58.3 = const(int, 1) $58.4 = _n_23 - $const58.3

The first thing to note is that this information is for advanced users as it refers to the Numba IR of the function being transformed. As an example, the expression

`a * a`

in the example source partly translates to the expression`$arg_out_var.17 = $expr_out_var.9 * $expr_out_var.9`

in the IR, this clearly cannot be hoisted out of`loop #0`

because it is not loop invariant! Whereas in`loop #3`

, the expression`$const58.3 = const(int, 1)`

comes from the source`b[j + 1]`

, the number`1`

is clearly a constant and so can be hoisted out of the loop.

See also